# The importance of the whole: topological data analysis for the network neuroscientist

arXiv: https://arxiv.org/abs/1806.05167 · HTML: https://mlsanity.com/paper/1806.05167/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Jun 2018.

## Built on by (most cited first, 11 of 11)

- [Inference of hyperedges and overlapping communities in hypergraphs](https://mlsanity.com/paper/2204.05646/) (2022)
- [Minimal Cycle Representatives in Persistent Homology using Linear Programming: an Empirical Study with User's Guide](https://mlsanity.com/paper/2105.07025/) (2021)
- [Topological Causal Effects](https://mlsanity.com/paper/2603.02289/) (2026)
- [Persistent Homological State-Space Estimation of Functional Human Brain Networks at Rest](https://mlsanity.com/paper/2201.00087/) (2022)
- [PERCEPT: a new online change-point detection method using topological data analysis](https://mlsanity.com/paper/2203.04246/) (2022)
- [A topological classifier to characterize brain states: When shape matters more than variance](https://mlsanity.com/paper/2303.04231/) (2023)
- [Chordless cycle filtrations for dimensionality detection in complex networks via topological data analysis](https://mlsanity.com/paper/2509.08350/) (2025)
- [A Class of Topological Pseudodistances for Fast Comparison of Persistence Diagrams](https://mlsanity.com/paper/2402.14489/) (2024)
- [Directed Semi-Simplicial Learning with Applications to Brain Activity Decoding](https://mlsanity.com/paper/2505.17939/) (2025)
- [Detecting Narrative Shifts through Persistent Structures: A Topological Analysis of Media Discourse](https://mlsanity.com/paper/2506.14836/) (2025)
- [Time Series Analysis of Spiking Neural Systems via Transfer Entropy and Directed Persistent Homology](https://mlsanity.com/paper/2508.19048/) (2025)

## Abstract (arXiv)

The application of network techniques to the analysis of neural data has greatly improved our ability to quantify and describe these rich interacting systems. Among many important contributions, networks have proven useful in identifying sets of node pairs that are densely connected and that collectively support brain function. Yet the restriction to pairwise interactions prevents us from realizing intrinsic topological features such as cavities within the interconnection structure that may be just as crucial for proper function. To detect and quantify these topological features we must turn to methods from algebraic topology that encode data as a simplicial complex built of sets of interacting nodes called simplices. On this substrate, we can then use the relations between simplices and higher-order connectivity to expose cavities within the complex, thereby summarizing its topological nature. Here we provide an introduction to persistent homology, a fundamental method from applied topology that builds a global descriptor of system structure by chronicling the evolution of cavities as we move through a combinatorial object such as a weighted network. We detail the underlying mathematics and perform demonstrative calculations on the mouse structural connectome, electrical and chemical synapses in textitC. elegans, and genomic interaction data. Finally we suggest avenues for future work and highlight new advances in mathematics that appear ready for use in revealing the architecture and function of neural systems.